Our definition of a spacetime \((M,g)\)  will follow [1]}. In particular, the manifold \(M\)  is always assumed to be smooth. A smooth spacetime is one where the metric \(g\)  is smooth, that is, its components \(g_{\mu \nu } = g(\partial _\mu , \partial _\nu )\)  are smooth functions with respect to any coordinates \((x^0, \cdots , x^n)\) . A continuous spacetime is one where the metric is continuous, that is, its components are continuous functions with respect to any coordinates.
